On the lower bound side, Kuhn et al. [1]} provide lower bounds of the form
\(\Omega (\log \Delta /\log \log \Delta )\)  and
\(\Omega (\sqrt{\log n /\log \log n})\)  for constant-approximation to MVC
and polylogarithmic-approximation to MDS in the \(\mathsf {LOCAL}\)  model.
Naturally, these lower bounds apply to the \(\mathsf {CONGEST}\)  model as well.
In the \(\mathsf {LOCAL}\)  model MVC, MDS, \(G^2\) -MVC and \(G^2\) -MDS can be solved deterministically in \(\operatorname{\text{{\rm poly}}}\log n\)  rounds even if one aims for \((1+\varepsilon )\) -approximations with \(\varepsilon =1/\operatorname{\text{{\rm poly}}}\log n\)  [2]}, [3]}.
